Watch the Reg animations. As soon as you see him reach into his pocket to pull out the mines, then simply move.
You have two choices, you move back, then move a minion up or wait for a creep to take them out.
You go offensive and move forward, hope that you don't have to retreat, or remember to stick to a side.
Mines are really only dangerous, if you decide that you can just sit there and slug it out.
If you know they are mines build, then keep moving around.
MoB is a little harder to deal with, but I have found it rarely makes too much of a dent, unless I am taking a beating.
The problem is that playing an offensive QoT against mines Reg is bad, you have to get in close, this is gonna mean you can't react to mines.
Playing a minion QoT is much easier. You keep your range and harass him, you know when the mines are coming as he will have to run in a bit to you, or the actual time they are flying out gives you enough time to move.
The trouble is that you picked the wrong build. Against a nova Reg you would have had better luck, but a mines Reg is bad news for that build.
QoT is not designed to be a DG killer, your Nova Reg buddy should have been backing you up, pushing him back.